73.2.3 problem 3

Internal problem ID [19802]
Book : Elementary Differential Equations. By R.L.E. Schwarzenberger. Chapman and Hall. London. First Edition (1969)
Section : Chapter 4. Autonomous systems. Exercises at page 69
Problem number : 3
Date solved : Thursday, October 02, 2025 at 04:43:57 PM
CAS classification : [[_Emden, _Fowler], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} t^{2} x^{\prime \prime }-2 x^{\prime } t +2 x&=0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 11

ode:=t^2*diff(diff(x(t),t),t)-2*t*diff(x(t),t)+2*x(t) = 0; 
dsolve(ode,x(t), singsol=all);
 

\[ x = t \left (c_2 t +c_1 \right ) \]

✓ Mathematica. Time used: 0.103 (sec). Leaf size: 133

ode=t^2*D[x[t],{t,2}]-2*D[x[t],t]+2*x[t]==0; 
ic={}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to 2^{-\frac {1}{2} i \left (\sqrt {7}-i\right )} t^{\frac {1}{2}-\frac {i \sqrt {7}}{2}} \left (c_2 t^{i \sqrt {7}} \operatorname {Hypergeometric1F1}\left (-\frac {1}{2}-\frac {i \sqrt {7}}{2},1-i \sqrt {7},-\frac {2}{t}\right )+2^{i \sqrt {7}} c_1 \operatorname {Hypergeometric1F1}\left (\frac {1}{2} i \left (i+\sqrt {7}\right ),1+i \sqrt {7},-\frac {2}{t}\right )\right ) \end{align*}

✓ Sympy. Time used: 0.090 (sec). Leaf size: 8

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(t**2*Derivative(x(t), (t, 2)) - 2*t*Derivative(x(t), t) + 2*x(t),0) 
ics = {} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = t \left (C_{1} + C_{2} t\right ) \]